Small Winding-Number Expansion: Vortex Solutions at Critical Coupling

TL;DR

This paper introduces a novel perturbative expansion method for vortex solutions in the Abelian-Higgs model at critical coupling, extending the winding number to real values and validating the approach through analytical and numerical tests.

Contribution

It proposes a new small winding-number expansion technique that accurately describes vortex solutions, including for large winding numbers, using Padé approximation.

Findings

Successfully reproduces scalar charge with high precision

Works well for large winding numbers

Validates the expansion with analytical and numerical methods

Abstract

We study an axially symmetric solution of a vortex in the Abelian-Higgs model at critical coupling in detail. Here we propose a new idea for a perturbative expansion of a solution, where the winding number of a vortex is naturally extended to be a real number and the solution is expanded with respect to it around its origin. We test this idea on three typical constants contained in the solution and confirm that this expansion works well with the help of the Pad\'e approximation. For instance, we analytically reproduce the value of the scalar charge of the vortex with an error of $O(10^{-6})$. This expansion is also powerful even for large winding numbers.